Goals from Sagi Burton, Andy Cooke and Ben Davies gave the visitors a decisive triumph.
The result saw Boston slip to defeat for the fifth time in six matches during which time they had pocketed just one point.
It was the visitors who struck first when, after five minutes, Burton got a glancing header on a curling free kick from the left by Davies and the ball drifted wide of keeper Andy Marriott.
Boston had earlier seen a shooting effort by Ernie Cooksey fly wildly off target.
Following their breakthrough goal, Shrewsbury exerted more pressure with Derek Asamoah giving the home defence a testing time but without a final end product.
The home side struggled to make any sort of initial impact and long balls from the defence failed to unduly ruffle the Shrews' rearguard well marshalled by Burton and Kelvin Langmead.
Boston carved out another fleeting chance to get on the score sheet, but debutant Albert Jarrett fired far too high with an ambitious 25-yard drive.
Shrewsbury again threatened on 31 minutes with Dave Edwards seeing a dipping drive from distance drop inches over the home bar.
Boston looked to find a way back into the reckoning and they went close on 45 minutes when Shane Nicholson's 25-yard free kick curled round the visitors' wall but keeper Chris MacKenzie went full stretch to push the ball away.
Boston went 2-0 down on the stroke of half time when Cooke accepted a pass from Stewart Drummond and curled a 22-yard drive past Marriott.
The early stages of the second half saw the visitors continue to impress and twice Marriott had to dash out to save bravely at the feet of the speedy Asamoah.
Boston conceded a third goal in the 66th minute on the back of another telling Shrews attack.
Cooke controlled well on the left and squared for Davies to drill in off the underside of the bar from 18 yards.
Another Nicholson free kick from distance on 77 minutes came close to giving Boston a crumb of consolation, but his curling effort was expertly clutched by MacKenzie.
Shrewsbury replied and almost scored a fourth, but Davies' long-range drive bounced away of a post in the afternoon's final moment of telling action.